Qubicle to blender for import to Unity - don't know if texture is actually applied?
The 3DView has several different shading modes ranging from bounding box to render preview. By default, in solid shading the objects are drawn in a plain colour and any textures are not visible, while they aren't displayed in the 3DView they are used when rendering and exporting. When you change to texture painting, the textures are displayed as that is the data you are working with.

Something that can catch you out is that you need to save the image files after painting as they are not saved when you save the blend file. The answer to this question has a script that will save the images when you save the blend file.

If you search for "blender texture painting" you can find many tutorials to get you started.